This special volume of the ESAIM Journal, Mathematical Modelling and Numerical Analysis,contains a collection of articles on probabilistic interpretations of some classes of nonlinear integro-differential equations.The selected contributions deal with a wide range of topics in applied probability theory and stochastic analysis, with applications in a variety of scientific disciplines, includingphysics, biology, fluid mechanics, molecular chemistry, financial mathematics and bayesian statistics. In this preface, we provide a brief presentation of the main contributions presented in this special volume. We have also included an introduction to classic probabilistic methods and a presentation of the more recent particle methods, with a synthetic picture of their mathematical foundations and their range of applications.

Numerical simulation of high frequency waves in highly heterogeneous media is a challenging problem. Resolving the fine structure of the wave field typically requires extremely small time steps and spatial meshes. We show that capturing macroscopic quantities of the wave field, such as the wave energy density, is achievable with much coarser discretizations. We obtain such a result using a time splitting algorithm that solves separately and successively propagation and scattering in the simplified regime of the parabolic wave equation in a random medium. The mathematical theory of the convergence and statistical properties of the algorithm is based on the analysis of the Wigner transforms in random media. Our results provide a step toward understanding time and space discretizations that are needed in order for the numerical algorithm to capture the correct macroscopic statistics of the wave energy density in a random medium.

A coupled finite/boundary element method to approximate the free vibration modes of an elastic structure containing an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of one of the most usual procedures in engineering practice: an added mass formulation, which is posed in terms of boundary integral equations. Piecewise linear continuous elements are used to discretize the solid displacements and the fluid-solid interface variables. Spectral convergence is proved and error estimates are settled for the approximate eigenfunctions and their corresponding vibration frequencies. Implementation issues are also discussed and numerical experiments are reported.

We analyze the accuracy and well-posedness of generalized impedanceboundary value problems in the framework of scattering problemsfrom unbounded highly absorbing media. We restrict ourselves in this first workto the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficultiesin the analysis for the generalized impedance boundary conditions, sinceclassical compactness arguments are no longer possible. Our new analysisis based on the use of Rellich-type estimates and boundedness of L2solution operators. We also discuss some numerical experimentsconcerning these boundary conditions.

In the case of an elastic strip we exhibit two properties ofdispersion curves λn,n ≥ 1, that were not pointed outpreviously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on ${\mathbb{R}}_{+}$ . The non monotonicity was an open question (see [2],for example) and, for the first time, we give a rigourous answer. Recall thecharacteristic property of the dispersion curves: {λn(p);n ≥ 1} isthe set of eigenvalues of A p , counted with their multiplicity. Theoperators Ap , $p\in{\mathbb{R}}$ , are the reduced operators deduced from the elasticoperator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relationD(p,λ) = 0 in a general framework, and not only for a homogeneous situation(in this last case the relation is explicit). Recall that a dispersionrelation isan implicit equation the solutions of which are eigenvalues of A p . The mainproperty of the function D that we build is the following one: themultiplicity of an eigenvalue λ of A p is equal to the multiplicity ithas as a root of D(p,λ) = 0. We give also some applications.

For the Stokes problem in a two- or three-dimensionalbounded domain, we propose a new mixed finite element discretization which relies ona nonconforming approximation of the velocity and a more accurate approximation of thepressure. We prove that the velocity and pressure discrete spaces are compatible, in thesense that they satisfy an inf-sup condition of Babuška and Brezzi type, and wederive some error estimates.

We consider a family of quadrilateral or hexahedral mixed hp-finite elements for an incompressible flow problem with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and an arbitrary bound.For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size,we prove the inf-sup condition uniformly with respect to the meshsize and the polynomial degree.

We analyze two numerical schemes of Euler type in time and C 0finite-element type with $\mathbb{P}_1$ -approximation in space forsolving a phase-field model of a binary alloy with thermalproperties. This model is written as a highly non-linear parabolicsystem with three unknowns: phase-field, solute concentration andtemperature, where the diffusion for the temperature and soluteconcentration may degenerate. The first scheme is nonlinear, unconditionally stableand convergent. The other scheme is linear but conditionally stableand convergent. A maximum principle is avoided in both schemes,using a truncation operator on the L 2 projection onto the $\mathbb{P}_0$ finite element for the discrete concentration. Inaddition, for the model when the heat conductivity and solutediffusion coefficients are constants, optimal error estimates forboth schemes are shown based on stability estimates.

The Boltzmann–Poisson system modeling the electron flow in semiconductorsis used to discuss the validity of the Child–Langmuir asymptotics.The scattering kernel is approximated by a simple relaxation time operator.The Child–Langmuir limit gives an approximation of the current-voltagecharacteristic curves by means of a scalingprocedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical comparisons between the simulation of theBoltzmann–Poisson system and the Child–Langmuir equations in testproblems.

We study the 3-D elasticity problem in the case of a non-symmetric heterogeneous rod. The asymptotic expansion of the solution is constructed. The coercitivity of the homogenized equation is proved. Estimates are derived for the difference between the truncated series and the exact solution.

We deal with boundary layers and quasi-neutral limits in the drift-diffusion equations. We first show that this limit is unique and determined by a system of two decoupled equations with given initial and boundary conditions. Then we establish the boundary layer equations and prove the existence and uniqueness of solutions with exponential decay. This yields a globally strong convergence (with respect to the domain) of the sequence of solutions and an optimal convergence rate $O(\varepsilon^\frac{1}{2})$ to the quasi-neutral limit in L 2.

In this paper, we discuss advancedthermostatting techniques for sampling molecular systems in the canonical ensemble.We first survey work on dynamical thermostatting methods, including the Nosé-Poincaré method, and generalized bath methods which introduce a more complicated extended model to obtain better ergodicity. We describe a general controlled temperature model, projective thermostatting molecular dynamics(PTMD) and demonstrate that it flexibly accommodates existing alternativethermostatting methods, such as Nosé-Poincaré, Nosé-Hoover(with or without chains), Bulgac-Kusnezov, or recursive Nosé-PoincaréChains. These schemes offer possible advantages for use incomputing thermodynamic quantities, and facilitate the developmentof multiple time-scale modelling and simulation techniques. Inaddition, PTMD advances a preliminary step toward therealization of true nonequilibrium motion for selected degrees offreedom, by shielding the variables of interest from the artificialeffect of thermostats. We discuss extension of the PTMD method for constant temperature and pressure models. Finally, we demonstrate schemes for simulating systems with an artificial temperature gradient, by enabling the use of two temperature baths within the PTMD framework.

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite volume methods induce large errors when approximated the convection-diffusion extracted system. To solve this difficulty, recent works propose a nonlinear projection scheme based on cancellation phenomenon of relevant dissipation rates of entropy. Unfortunately, such a property never holds in the present framework. The nonlinear projection procedures are thus extended.

Our purpose is to estimate numerically the influence of particles on the global viscosity of fluid–particle mixtures. Particles are supposed to rigid, and the surrounding fluid is newtonian. The motion of the mixture is computed directly, i.e. all the particle motions are computed explicitly. Apparent viscosity, based on the force exerted by the fluid on the sliding walls, is computed at each time step of the simulation.In order to perform long–time simulations and still control the solid fraction, we assume periodicity of the flow in the shear direction.Direct simulations are based on the so–called Arbitrary Lagrangian Eulerian approach, which we adapted to make it suitable to periodic domains.As a first step toward modelling of interacting red cells in the blood, we propose a simple model of circular particles submitted to an attractive force which tends to form aggregates.

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. Wereduce the original problem by a Fourier expansion in the angular variable to a countablefamily of two-dimensional problems. We decompose the meridian domain, assumed polygonal,in a finite number of rectangles and we discretize by a spectral method. Then we describethe main features of the mortar method and use the algorithm Strang Fix to improve theaccuracy of our discretization.

This study is mainly dedicated to the development and analysis ofnon-overlapping domain decomposition methods for solving continuous-pressurefinite element formulations of the Stokes problem. These methods have thefollowing special features. By keeping the equations and unknowns unchanged atthe cross points, that is, points shared by more than two subdomains, one caninterpret them as iterative solvers of the actual discrete problem directlyissued from the finite element scheme. In this way, the good stabilityproperties of continuous-pressure mixed finite element approximations of theStokes system are preserved. Estimates ensuring that each iteration can beperformed in a stable way as well as a proof of the convergence of theiterative process provide a theoretical background for the application of therelated solving procedure. Finally some numerical experiments are given todemonstrate the effectiveness of the approach, and particularly to compare itsefficiency with an adaptation to this framework of a standard FETI-DP method.

This is the second part of the paper for a Non-Newtonian flow. Dual combined Finite Element Methods are used to investigate the littleparameter-dependent problem arising in a nonliner three field version of the Stokes system for incompressible fluids, where the viscosity obeys a general law including the Carreau's law and the Power law. Certain parameter-independent error bounds are obtained which solved the problem proposed by Baranger in [4] in a unifying way. We also give somestable finite element spaces by exemplifying the abstract B-B inequality. The continuous approximation for the extra stress is achieved as a by-product of the new method.

We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite-dimensional space of finite dimension notnecessarily included into V. We give a series of realisticconditions on an error estimator that allows to conclude that themarking strategy of bulk type leads to the geometric convergenceof the adaptive algorithm. These conditions are then verified fordifferent concrete problems like convection-reaction-diffusionproblems approximated by a discontinuous Galerkin methodwith an estimator of residual type or obtained by equilibratedfluxes. Numerical tests that confirm the geometric convergence arepresented.

We are interested in a barotropic motion of the non-Newtonian bipolarfluids .We consider a specialcase where the stress tensor is expressed in the form ofpotentials depending on e ii and $(\frac{\partiale_{ij}}{\partial x_{k}})$ .We prove theasymptotic stability of the rest state under the assumptionof the regularity of the potential forces.